Inequalities Examples

Example 1

Graph the inequality 2x + 4y ≤ 1.

First we graph the line 2x + 4y = 1:

Then we pick a point on one side of the line to see whether that point should be included or not. A nice point we can use is (0, 0). If we decide it can't be included, we'll hang a "No (0, 0) Allowed" sign on the clubhouse door and tell everyone else in the treehouse not to acknowledge him.

When x and y are both zero, the left-hand side of the inequality is

2(0) + 4(0)

which is indeed less than or equal to 1. That means we do want the side of the line with (0, 0), so we shade in that side of the graph:

When an inequality is strict, we do the same thing as above except that now we don't want to include the points on the line. In fact, we don't want to do anything that might upset it, considering how strict it is. We are so over being grounded.

To show that we aren't including the points on the line, we draw a dashed line instead of a solid line.

Example 2

Graph the inequality x + 1 < y.

First we graph the line x + 1 = y using a dotted line since we have a strict inequality. No, we don't care if your line is dotted or dashed. As long as it isn't made up of a string of tiny hearts.

Next, we pick a point on one side of the line. Again, (0, 0) is a good one to use. When x and y are both zero, the inequality is not satisfied, so we shade in the area on the other side of the line:

Example 3

Find the inequality graphed below.

First we find the equation of the line, which is y = 4x -3.

Since we have a dashed line, we know that we want a strict inequality: either < or >. If we take a point (x, y) in the shaded area of the graph, the value of y is less than it would be if we moved up to the line. Apparently, y's currency is worth more above the line, in the "ritzier" part of town.

Therefore, the inequality we want is y< 4x -3.

To check, we'll try a couple of points. The point (5, 0) is in the shaded area and satisfies the inequality, which is good. The point (0, 0) is not in the shaded area, and does not satisfy the inequality, which is also good. There's more good here than in a box of Good 'n Plenty.